Q: Can't open iPhoto

These are your options:

1. Restore the iPhoto library from the most recent backup that predates the issue.

2. Repair or rebuild the library.

3. Scavenge the library with iPhoto Library Manager independently of iPhoto (Library ▹ Rebuild). Books, calendars, and slideshows won't be preserved.

You say you moved the library to the Desktop. Did you try to create a new one, and fail? If so, please do as follows.

This procedure is a diagnostic test. It won’t solve your problem. Don’t be disappointed when you find that nothing has changed after you complete it.
   
The purpose of this test is to determine whether the problem is localized to your user account. Enable guest logins* and log in as Guest. Don't use the Safari-only “Guest User” login created by “Find My Mac.”
   
While logged in as Guest, you won’t have access to any of your personal files or settings. Applications will behave as if you were running them for the first time. Don’t be alarmed by this; it’s normal. If you need any passwords or other personal data in order to complete the test, memorize, print, or write them down before you begin.
   
Test while logged in as Guest. Same problem?
     
After testing, log out of the guest account and, in your own account, disable it if you wish. Any files you created in the guest account will be deleted automatically when you log out of it.
  
*Note: If you’ve activated “Find My Mac” or FileVault in OS X 10.7 or later, then you can’t enable the Guest account. The "Guest User" login created by "Find My Mac" is not the same. Create a new account in which to test, and delete it, including its home folder, after testing.

I've tried downloading the Library upgrade (though this surely shouldn't be necessary) but to no avail.

Did you run the upgrader after downloading it? 

What version of iPhoto did you have before the move to Mt. Lion?  If it was iPhoto 5 or earlier then the Library Upgrade would be necessary.

OT

It gets installed in the Applications / Utilities folder. Go there and double click on it.

I don't know where the rest have gone and they don't appear to have been saved in any recent backup on my portable drive.

How do you know that?

Apply the two fixes below in order as needed on your library:

Fix #1

1 - launch iPhoto with the Command+Option keys held down and rebuild the library.

2 - run Option #4 to rebuild the database.

Fix #2

Using iPhoto Library Manager  to Rebuild Your iPhoto Library

1 - download iPhoto Library Manager and launch.

2 - click on the Add Library button and select the library you want to add in the selection window..

3 - Now that the library is listed in the left hand pane of iPLM, click on your library and go to the Library ➙ Rebuild Library menu option.

4 - In the next  window name the new library and select the location you want it to be placed.

5 - Click on the Create button.

Note: This creates a new library based on the LIbraryData.xml file in the library and will recover Events, Albums, keywords, titles and comments.  However, books, calendars, cards and slideshows will be lost. The original library will be left untouched for further attempts at fixing the problem or in case the rebuilt library is not satisfactory.